Theorem rnun 5145

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	|- ran ( A u. B ) = ( ran A u. ran B )

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 5142	. . . 4 |- `' ( A u. B ) = ( `' A u. `' B )
2	1	dmeqi 4932	. . 3 |- dom `' ( A u. B ) = dom ( `' A u. `' B )
3		dmun 4938	. . 3 |- dom ( `' A u. `' B ) = ( dom `' A u. dom `' B )
4	2, 3	eqtri 2252	. 2 |- dom `' ( A u. B ) = ( dom `' A u. dom `' B )
5		df-rn 4736	. 2 |- ran ( A u. B ) = dom `' ( A u. B )
6		df-rn 4736	. . 3 |- ran A = dom `' A
7		df-rn 4736	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 3359	. 2 |- ( ran A u. ran B ) = ( dom `' A u. dom `' B )
9	4, 5, 8	3eqtr4i 2262	1 |- ran ( A u. B ) = ( ran A u. ran B )

Syntax hints:   = wceq 1397   u. cun 3198   `'ccnv 4724   dom cdm 4725   ran crn 4726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by:  imaundi  5149  imaundir  5150  rnpropg  5216  fun  5508  foun  5602  fpr  5835  fprg  5836  sbthlemi6  7160  exmidfodomrlemim  7411